Prove the following identities.
$\tan\theta=\frac{\sin\theta}{\cos\theta}$

Verified step by step guidance

1. Start by recalling the definition of the tangent function in terms of sine and cosine: $\tan \theta =\frac{\sin \theta }{\cos \theta }$. This is a fundamental trigonometric identity.
2. To prove the identity $\tan \theta =\frac{\sin \theta }{\cos \theta }$, we need to express $\tan \theta$ in terms of $\sin \theta$ and $\cos \theta$.
3. The tangent of an angle $\theta$ in a right triangle is defined as the ratio of the opposite side to the adjacent side. In terms of the unit circle, this translates to $\tan \theta =\frac{y}{x}$, where $y=\sin \theta$ and $x=\cos \theta$.
4. Thus, substituting the unit circle definitions, we have $\tan \theta =\frac{\sin \theta }{\cos \theta }$, which matches the given identity.
5. Therefore, the identity $\tan \theta =\frac{\sin \theta }{\cos \theta }$ is proven using the definitions of sine, cosine, and tangent in the context of the unit circle.